A decomposition of continuity

Abstract

In 1922 Blumberg[1] introduced the notion of a real valued function on Euclidean space being densely approached at a point in its domain. Continuous functions satisfy this condition at each point of their domains. This concept was generalized by Ptak[7] in 1958 who used the term ’nearly continuous’, and by Husain[3] in 1966 under the name of ’almost continuity’. More recently, Mashhour et al. [5] have called this property of functions between arbitrary topological spaces ’precontinuity’. In this paper we define a new property of functions between topological spaces which is the dual of Blumberg’s original notion, in the sense that together they are equivalent to continuity. Thus we provide a new decomposition of continuity in Theorem 4 (iv) which is of some historical interest. In a recent paper [10] , Tong introduced the notion of an A–set in a topological space and the concept of A–continuity of functions between topological spaces. This enabled him to produce a new decomposition of continuity. In this paper we improve Tong’s decomposition result and provide a decomposition of A–continuity.